Eisenstein primes, critical values and global torsion

Neil Dummigan

Vol. 233 (2007), No. 2, 291–308

DOI: 10.2140/pjm.2007.233.291

Abstract

We consider congruences between Eisenstein series and cusp forms — of weight k, level N and character χ of conductor N — modulo large prime divisors of L(1 − k,χ⁻¹). We show that such primes occur in the order of a “global torsion” group attached to the cusp form f, and (under a certain hypothesis) also in the denominator of the algebraic part of the rightmost critical value L_f(k − 1). These occurrences are linked by the Bloch–Kato conjecture.

Keywords

modular form, L-function, Bloch–Kato conjecture

Mathematical Subject Classification 2000

Primary: 11F67, 14G10

Milestones

Received: 10 January 2007

Accepted: 18 May 2007

Published: 1 December 2007